1. Field of the Invention
The invention relates to a beamforming technique, more particularly to a beamformer using cascade multi-order factors, and a signal receiving system incorporating the same.
2. Description of the Related Art
Beamforming technology, in which a signal is multiplied with a complex weight so as to adjust magnitude and phase thereof, is used in smart antennas for both transmission and reception. Since beamforming is normally implemented using digital signal processing (DSP) techniques, the complex weight must be quantized, resulting in weight quantization error, which often affects beamforming performance and system stability (such as in terms of zeros), and hence degrades communication quality.
Referring to FIG. 1, a carrier signal from a transmitting end (not shown) enters a conventional smart antenna 8 at an arrival angle (θ) relative to a broadside of the conventional smart antenna 8. The conventional smart antenna 8 includes a linear array of a number (N) of isotropic antenna units with uniform spacing, where (N) is a positive integer. An array pattern function obtained by combining output signals of the isotropic antennas, 1, u1, u2, . . . , uN−1, with respective weights w0, w1, w2, . . . , wN−1, can be represented by the following equation:
      P    ⁡          (      u      )        =            ∑              n        =        0                    N        -        1              ⁢                  w        n            ⁢                        u          n                .            
Assuming that the array pattern function P(u) has a number (N−1) of first order zeros, z1, z2, . . . , zN−1, then the array pattern function P(u) can also be represented by the following equation:
      P    ⁡          (      u      )        =            w              N        -        1              ⁢                  ∏                  i          =          1                          N          -          1                    ⁢                          ⁢                        (                      u            -                          z              i                                )                .            Equations (1) and (2) below are partial derivatives of the array pattern function P(u) respectively with respect to a particular weight wn and a particular zero zi, i.e.,
                    ∂                  P          ⁡                      (            u            )                                      ∂                  w          n                      ⁢                  ⁢    and    ⁢                  ⁢                  ∂                  P          ⁡                      (            u            )                                      ∂                  z          i                      ,where n=0, 1, 2, . . . , N−1 and i=0, 1, 2, . . . , N−1. An expression of
      ∂          z      i            ∂          w      n      is obtained using Equations (1) and (2), and is shown in Equation (3).
                                          ∂                          P              ⁡                              (                u                )                                                          ∂                          w              n                                      =                  u          n                                    (        1        )            
                                          ∂                          P              ⁡                              (                u                )                                                          ∂                          z              i                                      =                              -                          w                              N                -                1                                              ⁢                                    ∏                                                k                  =                  1                                ,                                  k                  ≠                  i                                                            N                -                1                                      ⁢                                                  ⁢                          (                              u                -                                  z                  k                                            )                                                          (        2        )                                                      ∂                          z              i                                            ∂                          w              n                                      =                                                                              ∂                                      P                    ⁡                                          (                      u                      )                                                                                        ∂                                      w                    n                                                              ⁢                              |                                  u                  =                                      z                    i                                                                                                                        ∂                                      P                    ⁡                                          (                      u                      )                                                                                        ∂                                      z                    i                                                              ⁢                              |                                  u                  =                                      z                    i                                                                                =                                    -                                                (                                      z                    i                                    )                                n                                                                    w                                  N                  -                  1                                            ⁢                                                ∏                                                            k                      =                      1                                        ,                                          k                      ≠                      i                                                                            N                    -                    1                                                  ⁢                                                                  ⁢                                  (                                                            z                      i                                        -                                          z                      k                                                        )                                                                                        (        3        )            
As seen from Equation (3), changes in each weight wn affect all the zeros z1, z2, . . . , zN−1 of the array pattern function P(u) implemented by the conventional smart antenna 8. Such changes in the weights wn may arise when, for example, the weights wx,t are generated according to different quantization wordlengths.
A total displacement for a particular zero zi (i.e., a zero displacement Δzi) can be expressed as a sum of all zero shifts due to the quantization errors of all of the weights w0, w1, w2, . . . , wN−1, i.e.,
            Δ      ⁢                          ⁢              z        i              =                  ∑                  n          =          0                          N          -          1                    ⁢                                    ∂                          z              i                                            ∂                          w              n                                      ⁢        Δ        ⁢                                  ⁢                  w          n                      ,where i=0, 1, 2, . . . , N−1. By substituting Equation (3) into the above equation for the zero displacement Δzi, it can be obtained that
      Δ    ⁢                  ⁢          z      i        =            ∑              n        =        0                    N        -        1              ⁢                            -                                    (                              z                i                            )                        n                                                w                          N              -              1                                ⁢                                    ∏                                                k                  =                  1                                ,                                  k                  ≠                  i                                                            N                -                1                                      ⁢                                                  ⁢                          (                                                z                  i                                -                                  z                  k                                            )                                          ⁢      Δ      ⁢                          ⁢                        w          n                .            
Therefore, a quantitative measure (Qprior) for the effect of weight quantization error on the array pattern function P(u) implemented by the conventional smart antenna 8 can be defined by Equation (4) below:
                              Q          prior                =                                            ∑                              i                =                1                                            N                -                1                                      ⁢                                                        Δ                ⁢                                                                  ⁢                                  z                  i                                                                            =                                    ∑                              i                =                1                                            N                -                1                                      ⁢                                                                          ∑                                      n                    =                    0                                                        N                    -                    1                                                  ⁢                                                                                                    (                                                  z                          i                                                )                                            n                                                                                      w                                                  N                          -                          1                                                                    ⁢                                                                        ∏                                                                                    k                              =                              1                                                        ,                                                          k                              ≠                              i                                                                                                            N                            -                            1                                                                          ⁢                                                                                                  ⁢                                                  (                                                                                    z                              i                                                        -                                                          z                              k                                                                                )                                                                                                      ⁢                  Δ                  ⁢                                                                          ⁢                                      w                    n                                                                                                                        (        4        )            
From Equation (4), it is evident that, when the zeros z1˜zN−1 are clustered in the array pattern function P(u),
      ∏                  k        =        1            ,              k        ≠        i                    N      -      1        ⁢      (                  z        i            -              z        k              )  induces a huge variation on the quantitative measure (Qprior) for the effect of weight quantization error. Consequently, the zero displacement Δzi is highly sensitive to the weight quantization error Δwn, which adversely affects communication quality of the conventional smart antenna 8 such that the communication quality easily deviates from system requirements and specification.